login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A103519 a(1) = 1, a(n) = sum of n successive numbers starting with a(n-1) + 1. 2
1, 5, 21, 94, 485, 2931, 20545, 164396, 1479609, 14796145, 162757661, 1953092010, 25390196221, 355462747199, 5331941208105, 85311059329816, 1450288008607025, 26105184154926621, 495998498943605989, 9919969978872119990 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, the triangular series (1, 3, 6, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..449

FORMULA

a(n+1) = k(k+1)/2 - a(n)*(a(n)+1)/2, where k = a(n) + n +1.

a(n) = Sum_{i=0..n} (n!/(n-i)!) * (n-i)(n-i+1)/2 = Sum_{i=0..n} (n!/(n-i)!) * A000217(n-i). For n > 2, a(n) = (3*n*(n-1)/2)*floor((n-2)!*e) + n, where e=exp(1). - Max Alekseyev, Feb 14 2005

a(n) = n*a(n-1) + n*(n+1)/2. - Emeric Deutsch, Mar 16 2008

a(n) ~ 3*sqrt(Pi/2)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016

EXAMPLE

a(2) = 2 + 3 = 5, a(3) = 6 + 7 + 8 = 21, a(4) = 22 + 23 + 24 + 25.

MAPLE

a[1]:=1: for n from 2 to 20 do a[n]:=n*a[n-1]+(1/2)*n*(n+1) end do: seq(a[n], n=1..20); # Emeric Deutsch, Mar 16 2008

MATHEMATICA

RecurrenceTable[{a[1]==1, a[n]==n*a[n-1]+(n(n+1))/2}, a, {n, 20}] (* Harvey P. Dale, Nov 05 2013 *)

PROG

(PARI) { t(n) = n*(n+1)/2 } { a(n) = sum(i=0, n, n!/(n-i)!*t(n-i)) } { a2(n) = 3*t(n-1)*floor((n-2)!*exp(1))+n } \\ Max Alekseyev, Feb 14 2005

CROSSREFS

Cf. A103520.

Sequence in context: A116904 A126952 A273570 * A178876 A202513 A343349

Adjacent sequences: A103516 A103517 A103518 * A103520 A103521 A103522

KEYWORD

easy,nonn

AUTHOR

Amarnath Murthy, Feb 10 2005

EXTENSIONS

More terms from Max Alekseyev, Feb 14 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 17:17 EST 2022. Contains 358510 sequences. (Running on oeis4.)